1. Does there exist a positive integer which is a power of 2, such that we can obtain another power of 2 by rearranging it's digits?

2. Consider the triangle ABC with the angles B and C each larger than 45 degrees. We build the right-angled and isosceles triangles CAM and BAN outside the triangle ABC such that the right angles are and and BPC inside ABC such that it's right angle is . Prove that the triangle MNP is also right-angled and isosceles.

3. We have a 100 x 100 lattice with a tree on each of the 10000 points. (The points are equally spaced.) Find the maximum number of trees we can cut such that if we stand on any cut tree, we see no trees which have been cut. (In other words, on the line connecting any two trees that have been cut, there should be at least one tree which hasn't been cut.)

Second DayMay 6th, 1999Time : 4 hours

4. Find all natural numbers m such that :

m = 1/a1 +2/a2 + 3/a3 + ... + 1378/a1378

where a1 , ... , a1378 are natural numbers.

5. Consider the triangle ABC . P , Q and R are points on the sides AB , AC and BC respectively. A' , B' and C' are points on the lines PQ , PR and QR respectively such that ABllA'B' , ACllA'C' and BCllB'C' . Prove that AB/A'B' = SABC/SPQR (SABCmeans the area of the figure ABC)

6. A1 , A2 , ... , An are n distinct points on the plane. We color the middle of each line AiAj (i & j are not equal.) red. Find the minimum number of red points.

First DayApril 24th, 1997Time : 4 hours

1. x and y are two natural numbers such that 3x2 + x = 4y2 +y . Prove that x - y is the square of a whole nnumber.

2. Assume that KI and KN are the two tangents drawn from K onto the circle C . M is an arbitrary point on the extension of KN (near N) and P is the other extension point of the circle C with the circumcircle of KLM . Q is the foot of the altitude drawn from N onto ML . Prove that the measure of the angle MPQ is two times the angle KML .

3. Consider an n x n matrix of 0 , +1 and -1 , such that in each row and each column, there exists only one +1 and one -1 . Prove that by a finite number of changing columns with eachother and rows with eachother, we can change the places of +1's with -1's and vice versa.

5. In the triangle ABC , B and C are acute angles. The altitude of the triangle drawn from Aintersects BC at D . The bisectors of the angles B and C intersect AD at E and F respectively . If BE = CF , Prove that the triangle ABC is isosceles.

6. Find the largest p such that a and b are two natural numbers and p = b/4 ((2a-b)/(2a+b))1/2 is a prime number.